quiz9soln

quiz9soln - MATH 231 / Spring 2008 Name: Key 1. (4 points)...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 231 / Spring 2008 Name: Key 1. (4 points) Convert the point (x, y, z) = 1 1 1 2, 2, 2 April 2, 2008 into cylindrical and spherical. Solution: For cylindrical, we're already part of the way there: we know z. Thus, we only need 2 1 1 2 compute r and . We have an equation for r; namely, r2 = 1 + 2 = 1 , so r = 2 . Furthermore, 2 2 we know that = tan-1 (1) = /4. 1 1 Now for spherical. For spherical, we know that 2 = x2 + y 2 + z 2 = r2 + z 2 = 2 + 2 = 1. Also, we r already computed in our cylindrical, so we only need to compute . We know that = tan-1 ( z ) = -1 tan (1) = /4. f (x, y, z) dV as an iterated integral in 6 different ways, where E 2. (6 points) Express the integral E is the solid bounded by the surfaces x = 2, y = 2, z = 0, and x + y - 2z = 2. Solution: First, let's graph the region that we're looking at: 1.0 0.5 2.0 0.0 0.0 0.5 1.0 0.5 1.5 2.0 0.0 1.0 1.5 Note that I have set this up in a rather non-standard orientation for better viewing. The origin of this graph is located at the bottom left corner of the box. As such, the positive y axis moves up and right, whereas the positive x moves down and right. From here we can pretty much read off what our answers will be: 2 2 (x+y-2)/2 2 2 (x+y-2)/2 2 y/2 2 f (x, y, z) dz dy dx, 0 2-y 0 f (x, y, z) dz dx dy, 0 0 f (x, y, z) dx dz dy, 0 2-x 0 2-y+2z 1 2 2 2 x/2 2 1 2 2 f (x, y, z) dx dy dz, 0 0 f (x, y, z) dy dz dx, and 0 2z 2-x+2z f (x, y, z) dy dx dz. 0 2z 2-y+2z 2-x+2z 1 ...
View Full Document

Ask a homework question - tutors are online